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Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response

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  • Wang, Henan
  • Liu, Ping

Abstract

In this paper, a diffusive predator–prey model with Allee effect and functional response of generalized Holling type IV is established. The complex pattern dynamics of and bifurcation phenomena of system are analyzed by using linear stability analyses and bifurcation theory, etc. Corresponding to spiral pattern, spot pattern or spot–stripe mix pattern and chaotic pattern, respectively, the conditions that arise Hopf instability, Turing instability and Hopf–Turing instability of constant positive steady solutions of reaction–diffusion system are presented. The impacts of Allee effect and cross-diffusion on pattern formations of the system are further discussed. Particularly, provided that cross-diffusion is absent, there will not appear the formation of Turing pattern, that is, Turing pattern is only driven by cross-diffusions. Addressed to the different pattern formations involving in cross-diffusion and Allee effect, overall numerical simulations are provided when the spatial dimension is two and the theoretical results are demonstrated. It is revealed that the larger Allee effect constant A and the cross-diffusion coefficient d4 are advantageous for the appearing of formation of Turing pattern, while the cross-diffusion coefficient d3 suppresses the formation of Turing pattern with its increasing.

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  • Wang, Henan & Liu, Ping, 2023. "Pattern dynamics of a predator–prey system with cross-diffusion, Allee effect and generalized Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    • Handle: RePEc:eee:chsofr:v:171:y:2023:i:c:s0960077923003570
    DOI: 10.1016/j.chaos.2023.113456
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    References listed on IDEAS

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    1. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2023. "Qualitative study of cross-diffusion and pattern formation in Leslie–Gower predator–prey model with fear and Allee effects," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    2. Peng, Yahong & Zhang, Tonghua, 2016. "Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 1-12.
    3. Wang, Caiyun & Qi, Suying, 2018. "Spatial dynamics of a predator-prey system with cross diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 55-60.
    4. Peng, Yahong & Ling, Heyang, 2018. "Pattern formation in a ratio-dependent predator-prey model with cross-diffusion," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 307-318.
    5. Yan, Shuixian & Jia, Dongxue & Zhang, Tonghua & Yuan, Sanling, 2020. "Pattern dynamics in a diffusive predator-prey model with hunting cooperations," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
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    Cited by:

    1. Tang, Xiaosong & Zhang, Xiaoyu & Liu, Yiting & Li, Wankun & Zhong, Qi, 2023. "Global stability and Turing instability deduced by cross-diffusion in a delayed diffusive cooperative species model," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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